where \(H_\nu \)  are invertible linear operators, appropriately chosen so as to encode first-order information about \(R_\gamma \) . This is done by enforcing the inverse secant condition \(s^\nu = H_{\nu +1} y^\nu \) , for \(s^\nu =u^{\nu +1}-u^{\nu }\)  and \(y^\nu = R_\gamma (u^{\nu +1})-R_\gamma (u^{\nu })\)  and can be obtained by quasi-Newtonian methods such as the limited-memory BFGS (L-BFGS) method [1]} which is free from matrix operations, requires only a limited number of inner products and is suitable for embedded implementations.
The main weakness of this approach is that convergence is only guaranteed in a neighborhood of a \(\gamma \) -critical point \(u^\star \) . We shall describe a globalization procedure which hinges on the notion of the forward-backward envelope function.
